Positive Measure Spectrum for Schroedinger Operators with Periodic Magnetic Fields

We study Schroedinger operators with periodic magnetic field in Euclidean 2-space, in the case of irrational magnetic flux. Positive measure Cantor spectrum is generically expected in the presence of an electric potential. We show that, even without electric potential, the spectrum has positive measure if the magnetic field is a perturbation of a constant one.Comment: 17 page

An inviscid dyadic model of turbulence: the fixed point and Onsager's conjecture

Properties of an infinite system of nonlinearly coupled ordinary differential equations are discussed. This system models some properties present in the equations of motion for an inviscid fluid such as the skew symmetry and the 3-dimensional scaling of the quadratic nonlinearity. It is proved that the system with forcing has a unique equilibrium and that every solution blows up in finite time in $H^{5/6}$-norm. Onsager's conjecture is confirmed for the model system

On products of skew rotations

Let $H_1(p,q)$, $H_2(p,q)$ be two time-independent Hamiltonians with one degree of freedom and $\{S_1^t\}$, $\{S_2^t\}$ be the one-parametric groups of shifts along the orbits of Hamiltonian systems generated by $H_1$, $H_2$. In some problems of population genetics there appear the transformations of the plane having the form $T^{(h_1,h_2)}=S^{h_2}_2\cdot S_1^{h_1}$ under some conditions on $H_1$, $H_2$. We study in this paper asymptotical properties of trajectories of $T^{(h_1,h_2)}$.Comment: 13 pages, 10 figure

Space-Time Complexity in Hamiltonian Dynamics

New notions of the complexity function C(epsilon;t,s) and entropy function S(epsilon;t,s) are introduced to describe systems with nonzero or zero Lyapunov exponents or systems that exhibit strong intermittent behavior with ``flights'', trappings, weak mixing, etc. The important part of the new notions is the first appearance of epsilon-separation of initially close trajectories. The complexity function is similar to the propagator p(t0,x0;t,x) with a replacement of x by the natural lengths s of trajectories, and its introduction does not assume of the space-time independence in the process of evolution of the system. A special stress is done on the choice of variables and the replacement t by eta=ln(t), s by xi=ln(s) makes it possible to consider time-algebraic and space-algebraic complexity and some mixed cases. It is shown that for typical cases the entropy function S(epsilon;xi,eta) possesses invariants (alpha,beta) that describe the fractal dimensions of the space-time structures of trajectories. The invariants (alpha,beta) can be linked to the transport properties of the system, from one side, and to the Riemann invariants for simple waves, from the other side. This analog provides a new meaning for the transport exponent mu that can be considered as the speed of a Riemann wave in the log-phase space of the log-space-time variables. Some other applications of new notions are considered and numerical examples are presented.Comment: 27 pages, 6 figure

One-Dimensional Discrete Stark Hamiltonian and Resonance Scattering by Impurities

A one-dimensional discrete Stark Hamiltonian with a continuous electric field is constructed by extension theory methods. In absence of the impurities the model is proved to be exactly solvable, the spectrum is shown to be simple, continuous, filling the real axis; the eigenfunctions, the resolvent and the spectral measure are constructed explicitly. For this (unperturbed) system the resonance spectrum is shown to be empty. The model considering impurity in a single node is also constructed using the operator extension theory methods. The spectral analysis is performed and the dispersion equation for the resolvent singularities is obtained. The resonance spectrum is shown to contain infinite discrete set of resonances. One-to-one correspondence of the constructed Hamiltonian to some Lee-Friedrichs model is established.Comment: 20 pages, Latex, no figure

Isoperiodic deformations of the acoustic operator and periodic solutions of the Harry Dym equation

We consider the problem of describing the possible spectra of an acoustic operator with a periodic finite-gap density. We construct flows on the moduli space of algebraic Riemann surfaces that preserve the periods of the corresponding operator. By a suitable extension of the phase space, these equations can be written with quadratic irrationalities.Comment: 15 page

Phonon Localization in One-Dimensional Quasiperiodic Chains

Quasiperiodic long range order is intermediate between spatial periodicity and disorder, and the excitations in 1D quasiperiodic systems are believed to be transitional between extended and localized. These ideas are tested with a numerical analysis of two incommensurate 1D elastic chains: Frenkel-Kontorova (FK) and Lennard-Jones (LJ). The ground state configurations and the eigenfrequencies and eigenfunctions for harmonic excitations are determined. Aubry's "transition by breaking the analyticity" is observed in the ground state of each model, but the behavior of the excitations is qualitatively different. Phonon localization is observed for some modes in the LJ chain on both sides of the transition. The localization phenomenon apparently is decoupled from the distribution of eigenfrequencies since the spectrum changes from continuous to Cantor-set-like when the interaction parameters are varied to cross the analyticity--breaking transition. The eigenfunctions of the FK chain satisfy the "quasi-Bloch" theorem below the transition, but not above it, while only a subset of the eigenfunctions of the LJ chain satisfy the theorem.Comment: This is a revised version to appear in Physical Review B; includes additional and necessary clarifications and comments. 7 pages; requires revtex.sty v3.0, epsf.sty; includes 6 EPS figures. Postscript version also available at http://lifshitz.physics.wisc.edu/www/koltenbah/koltenbah_homepage.htm

Incommensurate Charge Density Waves in the adiabatic Hubbard-Holstein model

The adiabatic, Holstein-Hubbard model describes electrons on a chain with step $a$ interacting with themselves (with coupling $U$) and with a classical phonon field \f_x (with coupling \l). There is Peierls instability if the electronic ground state energy F(\f) as a functional of \f_x has a minimum which corresponds to a periodic function with period ${\pi\over p_F}$, where $p_F$ is the Fermi momentum. We consider ${p_F\over\pi a}$ irrational so that the CDW is {\it incommensurate} with the chain. We prove in a rigorous way in the spinless case, when \l,U are small and {U\over\l} large, that a)when the electronic interaction is attractive $U<0$ there is no Peierls instability b)when the interaction is repulsive $U>0$ there is Peierls instability in the sense that our convergent expansion for F(\f), truncated at the second order, has a minimum which corresponds to an analytical and ${\pi\over p_F}$ periodic \f_x. Such a minimum is found solving an infinite set of coupled self-consistent equations, one for each of the infinite Fourier modes of \f_x.Comment: 16 pages, 1 picture. To appear Phys. Rev.

A Nonperturbative Eliasson's Reducibility Theorem

This paper is concerned with discrete, one-dimensional Schr\"odinger operators with real analytic potentials and one Diophantine frequency. Using localization and duality we show that almost every point in the spectrum admits a quasi-periodic Bloch wave if the potential is smaller than a certain constant which does not depend on the precise Diophantine conditions. The associated first-order system, a quasi-periodic skew-product, is shown to be reducible for almost all values of the energy. This is a partial nonperturbative generalization of a reducibility theorem by Eliasson. We also extend nonperturbatively the genericity of Cantor spectrum for these Schr\"odinger operators. Finally we prove that in our setting, Cantor spectrum implies the existence of a $G_\delta$-set of energies whose Schr\"odinger cocycle is not reducible to constant coefficients